+ All Categories
Home > Documents > 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong...

2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong...

Date post: 19-Nov-2020
Category:
Upload: others
View: 5 times
Download: 0 times
Share this document with a friend
17
Appl. Sci. 2015, 5, 157-173; doi:10.3390/app5030157 applied sciences ISSN 2076-3417 www.mdpi.com/journal/applsci Article Prediction of Experimental Rainfall-Eroded Soil Area Based on S-Shaped Growth Curve Model Framework Wen Nie 1,2, *, Run-Qiu Huang 2,† , Qian-Gui Zhang 3 , Wei Xian 4 , Feng-Lin Xu 3,† and Lin Chen 5,† 1 State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing 400044, China 2 State Key Laboratory of Geo-hazard Prevention and Geo-environment Protection, Chengdu University of Technology, Chengdu 610059, Sichuan, China; E-Mail: [email protected] 3 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, 610500, Sichuan, China; E-Mails: [email protected] (Q.-G.Z.); [email protected] (F.-L.X.) 4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: [email protected] 5 School of Science, Southwest Petroleum University, Chengdu 610500, Sichuan, China; E-Mail: [email protected] These authors contributed equally to this work. * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +49-017631199092. Academic Editor: Takayoshi Kobayashi Received: 10 May 2015 / Accepted: 7 July 2015 / Published: 14 July 2015 Abstract: Rainfall-induced soil erosion of a mountain area plays a significant role in supplying sediment and shaping the landscape. The related area of soil erosion, as an index of the changed landscape, is easier to calculate visually using some popular imaging tools. By image analysis, our work shows that the changing of the soil erosion area admits the structure of an S-growth curve. Therefore, we propose to establish an S-curve model, based on incremental learning, to predict the soil erosion area. In the process of incremental learning, we dynamically update the accumulative rainfall and rainfall intensity to train the parameters of our S-curve model. In order to verify our prediction model, the index of area is utilized to express the output of eroded soil in a series of experiments. The results show that the proposed S-growth curve model can be used to estimate the growth of the soil OPEN ACCESS
Transcript
Page 1: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5, 157-173; doi:10.3390/app5030157

applied sciences ISSN 2076-3417

www.mdpi.com/journal/applsci

Article

Prediction of Experimental Rainfall-Eroded Soil Area Based on

S-Shaped Growth Curve Model Framework

Wen Nie 1,2,*, Run-Qiu Huang 2,†, Qian-Gui Zhang 3, Wei Xian 4, Feng-Lin Xu 3,† and

Lin Chen 5,†

1 State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University,

Chongqing 400044, China 2 State Key Laboratory of Geo-hazard Prevention and Geo-environment Protection,

Chengdu University of Technology, Chengdu 610059, Sichuan, China; E-Mail: [email protected] 3 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation,

Southwest Petroleum University, Chengdu, 610500, Sichuan, China;

E-Mails: [email protected] (Q.-G.Z.); [email protected] (F.-L.X.) 4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China;

E-Mail: [email protected] 5 School of Science, Southwest Petroleum University, Chengdu 610500, Sichuan, China;

E-Mail: [email protected]

† These authors contributed equally to this work.

* Author to whom correspondence should be addressed; E-Mail: [email protected];

Tel.: +49-017631199092.

Academic Editor: Takayoshi Kobayashi

Received: 10 May 2015 / Accepted: 7 July 2015 / Published: 14 July 2015

Abstract: Rainfall-induced soil erosion of a mountain area plays a significant role in

supplying sediment and shaping the landscape. The related area of soil erosion, as an index

of the changed landscape, is easier to calculate visually using some popular imaging tools.

By image analysis, our work shows that the changing of the soil erosion area admits the

structure of an S-growth curve. Therefore, we propose to establish an S-curve model, based

on incremental learning, to predict the soil erosion area. In the process of incremental

learning, we dynamically update the accumulative rainfall and rainfall intensity to train the

parameters of our S-curve model. In order to verify our prediction model, the index of area

is utilized to express the output of eroded soil in a series of experiments. The results show

that the proposed S-growth curve model can be used to estimate the growth of the soil

OPEN ACCESS

Page 2: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 158

erosion area (average relative error 3%–9.7%) according to variable soil material and

rainfall intensity. The original S-growth curve model can calculate the erosion areas of just

one soil material and one rainfall condition whose average relative error is 7.5%–12.2%;

compared to the simple time series analysis-moving average method (average relative error

5.7%–12.1%), our proposed S-growth curve model can reveal the physical mechanism and

evolution of the research object.

Keywords: eroded soil area; S-shaped growth curve; time series analysis;

incremental learning

1. Introduction

Rainfall-induced soil erosion involves three main processes: detachment, transport, and deposition

of soil particles by the crush forces of rainfall and surface runoff. During the erosion process, the area

of soil erosion can reflect the shaped landscape. The classical estimation models of soil erosion include

the USLE model, which is composed of six factors to predict the long-term average annual soil

weight [1,2]; the WEPP model, which is capable of predicting spatial and temporal distributions of soil

detachment and deposition on an event or continuous basis at both small (hillslopes, roads, small

parcels) and large (watershed) scales [3]; the EUROSEM model, which is a dynamic distributed

model, able to simulate sediment transport, erosion and deposition by rill and inter-rill processes in

single storms for both individual fields and small catchments [4]; the KINEROS model, which is an

event-oriented, physically-based model describing the processes of interception, infiltration, surface

runoff and erosion from agricultural and urban watersheds [5]. These models can commonly describe

the distributions of soil detachment in large (watershed) scales. By contrast, in our study, a narrow

scale of even a single slope is our research object. We aim to investigate the development of the area

of soil erosion induced by rainfall in a single slope using an imaging tool. In the practical experiments,

the values of rainfall intensity and cumulative rainfall are acquired incrementally with respect to time.

The area of soil erosion is achieved by the imaging tool under time domains. Then, in this work, based

on incremental learning, we develop an S-curve model to predict the soil erosion area. In detail, we

update the input values chronologically to train the parameters of the S-curve model. Five experiments

involving different materials and different rainfall intensities are employed to validate the model.

2. Physical Experiments and Results

2.1. Physical Models, Monitoring Devices, and Experimental Procedures

Schmidt et al. [6] highlight the advantage of physically-based models: using minimal historical data

to produce meaningful results. Thus, a trend of developing and testing physically-based soil erosion

models appears [7–11]. In our study, a slope model is constructed in a flume device made of Plexiglas

(Figure 1). The slope inclination is 34° and the material is from Fengdu Mountain, near to Yangtze

River Bank, Chongqing, China, whose surface soil can be easily eroded under heavy rainfall. The soil

was air-dried and crushed to different test samples having three particle size distributions, as shown in

Page 3: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 159

Figure 2. The soil sample was squeezed by a plate with a 120 N force to form the slope.

In order to reduce the “boundary effect”, we used a covering of polytetrafluoroethylene (PTFE) whose

friction coefficient is 0.04 on both sides of the flume. In Figure 1, m is the moisture transducer

(frequency domain sensor) [12] whose contact area with the soil is less than 20 mm2 (measuring range

0%–100%, resolution 0.1%, deviation ±2%); p is the pore water pressure transducer (diameter

3 cm, height 1.6 cm, measuring range ±10 kPa, deviation ±0.2%); d is a channel for drainage;

one high-definition digital video camera (5 million pixels, position in Figure 1) is used to record the

changes of the eroded soil area (front scene). Every test is conducted under similar initial geometry,

and pre-added water content (deviation ±3%, average value ~5.2%). Persistent rainfall is averagely

simulated by three nozzle spray heads (~0.6 m high to slope surface) in five rainfall events. For soil

Sample 1, we used three rainfall intensity levels in three experiments (approximately 25, 45, and

65 mm/h, respectively) to investigate the effect of different rainfall intensities on the same material.

Then we focused on the effect of the different materials (Sample 1, Sample 2, and Sample 3), by

comparing their responses in experiments with a common rainfall intensity of 45 mm/h.

70cm

Camera

70cm

Camera

Figure 1. Geometry of physical model.

0

10

20

30

40

50

60

70

80

90

100

0.01 0.1 1 10Particle size(mm)

Perc

enta

ge p

assi

ng(%

)

Soil sample1Soil sample2Soil sample3

Figure 2. Particle-size distribution curves.

Page 4: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 160

2.2. Soil Erosion Area Calculation

A fast way to calculate a two-dimensional area of an object is to use the “BOUNDARY” command

in the AutoCAD software (Autodesk, Inc., San Rafael, CA, United States of America) [13]. Using

“BOUNDARY”, a point can be picked within the area to create a closed poly line or polygon domain.

The “Features” palette can be used to find the area and perimeter of the poly line or polygon domain.

By using the “MEASUREGEOM” and “AREA” commands, a series of points or a selected object is

specified to calculate the area. In Figure 3a, a coordinate is first established. Then the contour of the

eroded soil area is extracted in Figure 3b. Finally, in Figure 3c, calculation of the selected area is

executed by AutoCAD. In every experiment, the video data per minute is chronologically intercepted

to calculate the area. Lens distortions and the distance between camera and ground affect the accuracy

of the picture area. An error calculation is carried out in Figure 3a, and it is found that the blue part,

which concentrates on the upper part is a picture area error of ~4.2%, while most of the soil erosion

happens in the lower part, which means that the area error of soil erosion could be less than 4.2%.

y=130cm

x=55cm

error~4.2%

y=130cm

x=55cm

y=130cm

x=55cm

error~4.2%

(a) (b) (c)

Figure 3. Area calculation by AutoCAD (soil Sample 1 and rainfall intensity-65 mm/h):

(a) establishment of coordinate; (b) extraction of contour; and (c) calculation of areas.

2.3. Soil Erosion Area

Increased detachability produced by surface runoff shear forces and the impact of raindrops is

considered as the main force driving erosion [14]. Figure 4 illustrates that the toe of the slope is more

eroded than other positions, in consideration of more ground water supply from the upper parts. The

erosion of the top part of the slope without ground water supply from other parts shows a slow increase.

Thus, this is typical progressive soil erosion. Figure 5 shows the erosion of different materials under

the same time domain and rainfall intensity. It is found that the coarse particles (such as soil Sample 3)

have lower erosion than fine particles (such as soil Sample 1) because of their higher permeability and

stronger framework.

Page 5: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 161

(a) (b) (c) (d)

Figure 4. Different time domains in Sample 1 and rainfall intensity-65 mm/h test:

(a) t = 6 min; (b) t = 19 min; (c) t = 45 min; and (d) t = 70 min.

(a) (b) (c)

Figure 5. Different materials under the same time domain (45 mm/h; t = 52 min):

(a) soil Sample 1 and erosion area-771.21 mm2; (b) soil Sample 2 and erosion

area-421.31 mm2; (c) soil Sample 3 and erosion area-359.46 mm2.

In Figure 6, soil Sample 1 (25 mm/h) has the most obvious initial phase (threshold of soil erosion

action) because of the low rainfall intensity. By comparison, soil Sample 1 (45 mm/h) and soil

Sample 1 (65 mm/h) have fewer time lags. In the second acceleration stage, the erosion under higher

rainfall intensity obviously occurs faster because of the greater surface runoff production. In the last

phase, for one thing the coarse particles remain and the loss of fine particles makes the soil structure

stronger; for another, the water erosion develops towards the deep direction. Therefore, the soil erosion

area has a maximum value, but the volume could still increase. By contrast, for different materials,

soil Sample 1 (more fine particles), with the weakest structure, has the earliest start point of the

second erosion phase and a high erosion rate during the second stage because more surface fine

particles are lost. Until the final stage (relatively stable structure), the erosion rate is still higher than

for the coarse materials, but only towards the deep direction. For the area of soil erosion in the final

stage, the values of the three materials can be close, while for the erosion volume or weight the coarse

material has a lower value. This is because coarse particles like soil Sample 3 have a stronger structure

(high friction) and higher infiltration rate (producing less surface runoff) than fine particles like soil

Page 6: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 162

Sample 1. In addition, it can also be seen that the stronger the structure of coarse material against soil

erosion is, the greater is the time lag of the beginning and accomplishment of the erosion area compared to

fine material. In summary, the soil erosion is decided by both rainfall and the material’s strength.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 20 40 60 80 100 120 140 160 180 200

Time(min)

Are

a(m

m2)

soil sample1-25mm/hr soil sample1-45mm/hr soil sample1-65mm/hr

soil sample2-45mm/hr soil sample3-45mm/hr

Figure 6. Results of soil erosion experiments.

2.4. Hydrological Characteristics

Subsurface water content and soil pore water pressure might reflect the effects of rainfall on the

processes of surface sealing, runoff generation and soil sediment production. For one thing, high soil

moisture content or low negative pore water pressure results in an increased detachability by surface

runoff shear forces and raindrop impact [14]. For another aspect, a high water content of the soil might

decrease aggregate slaking and breakdown due to less air escaping upward, reducing the detachment of

soil by raindrop impact [15,16]. In Figure 1, m1 is at the middle deep position; m2 is near to the

surface; and m3 is at the low part. By contrast, p1 is at the middle depth position; p2 is near to the

surface; and p3 is at the toe of the slope. Figure 7a demonstrates that m2 near to the surface increases

more easily. Due to their depth, m1 and m3 have a certain time lag before increasing of water content.

Furthermore, m3 has a higher value and a shorter time lag because of the water flow supply.

In Figure 7b, the pore water pressure follows a similar pattern. Point p2, at the surface, is always

saturated to some degree and has no potential to increase. Point p3 has a higher value than p1 because

of its deeper position. For the same material, a higher rainfall intensity means more water input.

Therefore, m and p have higher values and lower time lags in Figures 8 and 9. For the different

materials, in Figures 8, 10 and 11 the coarse particles in soil Sample 3 have higher permeability or a

higher infiltration rate than fine particles. Thus, the time lag of pore water pressure production in

coarse materials is shorter and the value is higher. The moisture value depends on the balance of the

water infiltration, supply, and drainage.

Page 7: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 163

0

2

4

6

8

10

12

14

1 13 25 37 49 61 73 85 97 109 121 133 145 157 169

Time(min)

Mo

istu

re(%

)

m1m2m3

0

0.1

0.2

0.3

0.4

0.5

0.6

1 13 25 37 49 61 73 85 97 109 121 133 145 157 169

Time(min)

Po

re w

ate

r p

ress

ure

(kP

a)

p1p2p3

(a) (b)

Figure 7. Soil Sample 1-25 mm/h: (a) Moisture content of m1, m2, and m3; and

(b) pore water pressure of p1, p2, and p3.

0

2

4

6

8

10

12

14

16

18

0 25 50 75 100 125 150 175 200

Time(min)

Mo

istu

re(%

)

m1m2m3

-4

-3

-2

-1

0

1

2

0 25 50 75 100 125 150 175 200

Time(min)

Po

re w

ate

r p

ress

ure

(kP

a)

p1p2p3

(a) (b)

Figure 8. Soil Sample 1-45 mm/h: (a) Moisture content of m1, m2, and m3; and

(b) pore water pressure of p1, p2, and p3.

0

2

4

6

8

10

12

14

16

1 13 25 37 49 61 73 85 97

Time(min)

Mo

istu

re(%

)

m1m2m3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 13 25 37 49 61 73 85 97

Time(min)

Po

re w

ate

r p

ress

ure

(kP

a)

p1p2p3

(a) (b)

Figure 9. Soil Sample 1-65 mm/h: (a) Moisture content of m1, m2, and m3; and

(b) pore water pressure of p1, p2, and p3.

Page 8: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 164

0

2

4

6

8

10

12

14

16

1 13 25 37 49 61 73 85 97 109 121 133 145 157

Time(min)

Mo

istu

re(%

)

m1m2m3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 13 25 37 49 61 73 85 97 109 121 133 145 157

Time(min)

Po

re w

ate

r p

ress

ure

(kP

a)

p1p2p3

(a) (b)

Figure 10. Soil Sample 2-45 mm/h: (a) Moisture content of m1, m2, and m3; and

(b) pore water pressure of p1, p2, and p3.

0

2

4

6

8

10

12

14

16

1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193

Time(min)

Mo

istu

re(%

)

m1m2m3

0

0.5

1

1.5

2

2.5

1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193

Time(min)

Po

re w

ate

r p

ress

ure

(kP

a)

p1p2p3

(a) (b)

Figure 11. Soil Sample 3-45 mm/h: (a) Moisture content of m1, m2, and m3; and

(b) pore water pressure of p1, p2, and p3.

3. Soil Erosion Area Prediction Model

3.1. S-Shaped Growth Curve

The S-curve is one of the most common phenomena in nature: It is seen in the spreading of

populations, tumors, contaminants, innovations, and economic activity [17–20]. The pattern of an

S-shaped growth curve (Pearl growth curve) usually comprises three phases in an environment: the

object initially increases slowly; then it enters a positive acceleration phase, increasing rapidly and

approaching an exponential growth rate as in the J-shaped curve; and finally, the object declines in a

negative acceleration phase until reaching almost zero growth rate (logistic function).

y =

k

1+ be-ax,a > 0,k > 0,x > 0 (1)

As shown in Equation (1), this slowing of the growth rate reflects increasing environmental

resistance, which becomes proportionately more important at higher object densities. The point of

Page 9: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 165

stabilization, or zero growth rate, is termed the “saturation value” (symbolized by k) or “carrying

capacity” of the environment. a and b are the parameters that decide the trend of curve development;

y is the output object; and x is the input variable. It is usually summarized mathematically by the

logistic equation. In our case, when the cumulative rainfall (x) achieves a certain threshold, erosion

begins to occur. Thus, in the first stage, the eroded area (y) increases slowly, even having some time

lag, especially under low rainfall intensity; in the second phase, the eroded area increases swiftly due

to the shear force produced by surface runoff; in the final phase, because of surface loss of weak

particles while strong particles remain, the erosion develops deeper into the slope (“carrying capacity”

of the environment (k)), and the eroded area growth rate then begins to reduce, even to zero.

3.2. S-Shaped Growth Curve Fitting

According to Equation (1), the S-curves of the five experiments are fitted. In Equation (1), x is

cumulative rainfall; y is the eroded area; k is the maximum eroded area; a is a parameter related with

material; and b is a parameter to control the erosion rate. Figure 12 shows that the S-curve model

matches the soil erosion area development with a relative average error of 7.5%–12.2%. Table 1

presents the parameters in the five models.

0

500

1000

1500

2000

2500

3000

3500

0 500 1000 1500 2000 2500 3000 3500 4000 4500Cumulative rainfall (mm)

Are

a(m

m2)

Fitting

Observation

0

1000

2000

3000

4000

5000

6000

7000

0 800 1600 2400 3200 4000 4800 5600 6400Cumulative rainfall(mm)

Are

a(m

m2)

Fitting

Observation

(a) (b)

0

1000

2000

3000

4000

5000

6000

7000

0 600 1200 1800 2400 3000 3600 4200 4800Cumulative rainfall (mm)

Are

a(m

m2)

Fitting

Observation

0

1000

2000

3000

4000

5000

6000

7000

0 500 1000 1500 2000 2500 3000 3500 4000 4500Cumulative rainfall (mm)

Are

a(m

m2)

Fitting

Observation

(c) (d)

Figure 12. Cont.

Page 10: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 166

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 800 1600 2400 3200 4000 4800 5600 6400Cumulative rainfall(mm)

Are

a(m

m2)

Fitting

Observation

(e)

Figure 12. S-growth curve fitting: (a) soil Sample 1-25 mm/h; (b) soil

Sample 1-45 mm/h; (c) soil Sample 1-65 mm/h; (d) soil Sample 2-45 mm/h; and

(e) soil Sample 3-45 mm/h.

Table 1. Parameters of five S-shaped growth curves.

Name k a b

Soil Sample 1-25 mm/h 3000 1.088 × e4 10.4

Soil Sample 1-45 mm/h 6350 4.75 × e5 15.2

Soil Sample 1-65 mm/h 9250 4.215 × e5 1.176

Soil Sample 2-45 mm/h 6350 4.4 × e5 20.6

Soil Sample 3-45 mm/h 6350 4.267 × e5 16.9

3.3. Development of Prediction Model

In order to predict the S-shaped growth curve framework based on different soil materials under

different rainfall intensities, we used the previous observation values to train the parameters of a model

to predict the object in the next time domain and then repeat the process in the following time domain,

which is similar to the dynamically updated method. The modified S-shaped growth curve framework

is used to predict the eroded area 1ts at time t+1, as in Equation (2):

11 ( )

', 0, ' 0, 0

1

t t tt t ta x x

t

ks a k b

b e (2)

where ta , related to material, is the parameter affecting the erosion rate at time t. It is considered a

variable as time elapses because the particles of the soil change. tb is considered as the variable which

depends on the cumulative eroded area and rainfall intensity at time t. tx and 1tx are the cumulative

rainfall at time t and t−1, respectively.

In Equation (1), the initial value of 0b can be calculated when t is zero. Therefore,

b0= (

k '

s0

-1) (3)

where k’ is the “saturation value” or “carrying capacity” of the eroded area. From the results in Table

1, it is found that k is mainly affected by rainfall intensity m linearly, rather than the material parameter

a, which is read as

Page 11: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 167

' , 3.44 k λm μ m (4)

where m is the rainfall intensity and the coefficients λ (147.5) and μ (507.5) of a linear relationship

can be obtained by experimental data fitting. The minimum value of m is 3.44, meaning the threshold

of erosion beginning (in order to make k′ positive). If the detachment capacity is significantly lower

than the transport capacity, the process is referred to as “detachment-limited erosion” [21]. Materials

are transported only after being detached. Thus, it should have a threshold of the rainfall amount,

which is the start point of soil erosion. This agrees with the fact that the shear force induced by

runoff over the resistance force is a precondition of soil loss, as reported by The Water Erosion

Prediction Project [22] and SHETRAN [23]. The final model is as follows after putting Equations (3)

and (4) into Equation (5):

1

1( )

, 0, ' 0, 0

1 ( 1)

t t t

t t ta x x

t

λm μs a k s

λm μe

s

(5)

For the proposed model framework, we just need to input the rainfall intensity and the calculated

cumulative rainfall at time t without other parameters in order to obtain the eroded soil area at time

t+1. When new data at time t+1 is inputted, this model can predict the eroded soil area at time t+2.

Figure 13 indicates the five predictions of the S-growth curve model based on the physical

experiments. The prediction results match the observation values well and the average errors are

between 3% and 9.7%. The standard deviations Ns of the five time predictions are 1180, 1170, 1172,

1181, and 1174, respectively, according to Equation (6):

2

1

1( )

N

N ii

s x xN

(6)

Where ix is the observed value of the sample item and x is the mean value of these observations,

while the denominator N stands for the size of the sample: this is the square root of the sample variance,

which is the average of the squared deviations about the sample mean.

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

Cumulative rainfall(mm)

Are

a(m

m2)

Prediction

Observation

0 2000 4000 6000 8000

0

1000

2000

3000

4000

5000

6000

7000

Cumulative rainfall(mm)

Are

a(m

m2)

Prediction

Observation

(a) (b)

Figure 13. Cont.

Page 12: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 168

0 2000 4000 6000 8000 100000

2000

4000

6000

8000

10000

Cumulative rainfall(mm)

Are

a(m

m2)

Prediction

Observation

0 1000 2000 3000 4000 5000

0

1000

2000

3000

4000

5000

6000

7000

Cumulative rainfall(mm)

Are

a(m

m2)

Prediction

Observation

(c) (d)

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

Cumulative rainfall(mm)

Are

a(m

m2)

Prediction

Observation

(e)

Figure 13. Prediction of model: (a) soil Sample 1-25 mm/h (n = 174); (b) soil

Sample 1-45 mm/h (n = 171); (c) soil Sample 1-65 mm/h (n = 142); (d) soil

Sample 2-45 mm/h (n = 107); and (e) soil Sample 3-45 mm/h (n = 177).

4. Discussion

4.1. Dimensionless Quantities of Soil Erosion Area

The developments of the soil erosion area are made as dimensionless plots as shown in Figure 14.

For different rainfall intensities, heavy rainfall still has a higher erosion rate; in the second phase, the

erosion rate from the light rainfall is greater than that induced by heavy rainfall; in the final stage, all

the soil erosion rates have similar trends. In other words, the soil erosion has a first low and then high

sensitivity to low rainfall intensity; the soil erosion has a first high and then low sensitivity to high

rainfall intensity; in the last stage, they have a similar sensitivity. For different materials, the erosion

rate of weak material (Sample 1) is higher than strong material in the first stage; in the second and

third phases, there is almost no difference between materials. Thus, it can be considered that the

sensitivity of the soil erosion area to material type is not obvious.

Page 13: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 169

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cumulative rainfall

Are

a

soil sample1-25mm/hr soil sample1-45mm/hr

soil sample1-65mm/hr

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cumulative rainfall

Are

a

soil sample1-25mm/hrsoil sample1-45mm/hrsoil sample1-65mm/hr

(a) (b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cumulative rainfall

Are

a

soil sample1-45mm/hr soil sample2-45mm/hr

soil sample3-45mm/hr

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cumulative rainfall

Are

a

soil sample1-45mm/hrsoil sample2-45mm/hrsoil sample3-45mm/hr

(c) (d)

Figure 14. Dimensionless plots of soil erosion area: (a) Soil erosion under different

rainfall intensity conditions; (b) soil erosion under different rainfall intensity conditions

(fitting curve); (c) soil erosion under different material conditions; and (d) soil erosion

under different material conditions (fitting curve).

4.2. Comparison Between S-Curve Based Model and Time Series Analysis–Moving Average Model

A case is investigated using a simple time series analysis–moving average method to predict the

eroded soil area in Equation (7) [24]. This method in Equation (7) uses the average of the previous

three values as the prediction value.

1 2 3' , 43

t t t

t

s s ss t (7)

where 'ts is the output object of the prediction model at time t. 1ts , 2ts , and 3ts are the observation

data at time t−1, t−2, and t−3, respectively. The result is as shown in Figure 15.

Page 14: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 170

0 1000 2000 3000 40000

500

1000

1500

2000

2500

3000

Cumulative rainfall(mm)A

rea

(mm

2)

Prediction

Observation

Figure 15. Prediction of moving averages method (Sample 1-25 mm/h) (n = 174).

Firstly, we did relative error analyses for the S-curve prediction model and moving averages model,

as shown in Figure 16 (for the case of sample1 under 25 mm/h rainfall). The average relative error of

all the experiments is 3%–9.7% for the S-curve prediction model. For the moving averages model, the

experimental average relative error is 5.7%–12.1%. The S-curve prediction model includes

information on the S-curve trend, and can thus describe the whole process, especially the changes of

phase. On the other hand, the moving averages model, whose prediction depends on the previous data,

cannot simulate the sudden change of Phase 1 to Phase 2. Thus, in the second stage, error exists all the

time until the final stage, reducing the increase of area. It has to be pointed out that the prediction

accuracy of the model still depends on the number of predictions. The greater the number of

predictions from the model, the lower its accuracy is, while the cost of accurate prediction is a smaller

number of forecasts. Secondly, the moving averages model is like a “black box”. Its use only involves

giving the input and getting the output without any clear physical meaning and process. In our study,

the S-growth curve model framework needs the rainfall intensity and cumulative rainfall (which can be

obtained from the rainfall intensity and time) as parameters. It can also describe the three phases of

eroded soil area growth from the physical angle. In addition, the model considers different soil

materials affecting the erosion area.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

25 325 625 925 1225 1525 1825 2125 2425 2725 3025 3325 3625 3925 4225

Cumulative rainfall(mm)

Re

lative

err

or

S-curve prediction model

Moving averages

Figure 16. Relative error analyses of S-curve prediction model and moving averages

model (Sample 1-25 mm/h).

Page 15: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 171

4.3. Limitation and Suggestion of Experiments

In the current stage, we did not consider the vegetation cover, slope effect and coupled with the

rainfall, material effect, because experiments are costly and very time-consuming. Slope shape and

surface roughness are relatively simple. As concerns the dimension effect, this is very common for any

mini-physical model experiment. In our case, the dimension effect arises mainly from the

underestimated rainfall impact and the “boundary effect”. The rainfall bump is reduced by the use of a

nozzle spray head in our experiment (we have revised this in Section 2.1). In order to reduce the

“boundary effect”, we use a covering of the polytetrafluoroethylene (PTFE), whose friction coefficient

is 0.04, on both sides of the flume. In the future, the impact on the soil erosion of the slope effect and

coupled the rainfall, slope, and material effect could be our next research direction.

5. Conclusions

In our work, we constructed a physical slope model to investigate to what extent different rainfall

intensities and slope materials affect the area of soil erosion acquired by an imaging tool. We further

presented an S-curve model to predict the growth of the eroded soil area, which is well trained

incrementally and verified efficiently by five physical experiments. The valuable conclusions include:

(1) The area of soil erosion obeys a trend of an S-growth curve under continuous rainfall.

(2) A higher rainfall intensity can produce a greater area of soil erosion than lower rainfall intensity

for the same material.

(3) Under the same rainfall intensity, coarse material (stronger material) has a time lag against the

soil erosion, but finally a similar area of soil erosion compared to fine material.

(4) The S-curve model coupled with incremental parameter learning can effectively predict the soil

erosion with just the rainfall information as input.

Acknowledgments

This research is supported by the Opening Fund of the State Key Laboratory of Geo-Hazard

Prevention and Geo-Environment Protection (Chengdu University of Technology)-SKLGP2013K007.

We are also grateful for support from the National Natural Science Foundation of China (51304170),

Young Scholars Development Fund of SWPU (201231010031).

Author Contributions

Drafting of manuscript: Wen Nie and Xian Wei; acquisition of data: Feng-Lin Xu; analysis and

interpretation of data: Qian-Gui Zhang and Wen Nie; Model construction: Xian Wei, Lin Chen,

Feng-Lin Xu and Wen Nie; and planning and supervision of the research: Run-Qiu Huang.

Page 16: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 172

Conflicts of Interest

The authors declare no conflict of interest.

References

1. Wischmeier, W.H.; Smith, D.D. Predicting Rainfall Erosion Losses; USDA Agricultural

Handbook, No. 537; U.S. Department of Agriculture: Washington, DC, USA, 1978.

2. Renard, K.G.; Foster, G.R.; Weesies, G.A.; McCool, D.K.; Yoder, D.C. Predicting Soil Erosion

by Water: A Guide to Conservation Planning with the Revised Universal Soil Loss Equation

RUSLE; Agriculture Handbook, No. 703; U.S. Department of Agriculture: Washington, DC, USA,

1997; p. 404.

3. Flanagan, D.C.; Nearing, M.A. USDA Water Erosion Prediction Project: Hillslope Profile and

Watershed Model Documentation; NSERL report, No. 10; USDA-ARS National Soil Erosion

Research Laboratory: West Lafayette, IN, USA, 1995.

4. Morgan, R.P.C.; Quinton, J.N.; Smith, R.E.; Govers, G.; Poesen, J.W.A.; Auerswald, K.;

Chisci, G.; Torri, D.; Styczen, M.E. The European soil erosion model EUROSEM: A dynamic

approach for predicting sediment transport from fields and small catchments. Earth Surf. Process.

Landf. 1993, 23, 527–544.

5. Smith, R.E.; Goodrich, D.C.; Quinton, J.N. Dynamic, distributed simulation of watershed erosion:

the KINEROS2 and EUROSEM models. J. Soil Water Conserv. 1995, 50, 517–520.

6. Soil Erosion: Application of Physically Based Models; Schmidt, J., Ed.; Springer Science &

Business Media: Berlin, Germany, 2013.

7. Nearing, M.A. Capabilities and limitations of erosion models and data. In Proceedings of the 13th

International Soil Conservation Organization Conference, Brisbane, Australia, 4–8 July 2004.

8. Pieri, L.; Bittelli, M.; Wu, J.Q.; Dun, S.; Flanagan, D.C.; Pisa, P.R.; Ventura, F.; Salvatorelli, F.

Using the Water Erosion Prediction Project (WEPP) model to simulate field observed runoff and

erosion in the Apennines mountain range, Italy. J. Hydrol. 2007, 336, 84–97.

9. Cerdà, A. Effects of rock fragment cover on soil infiltration, interrill runoff and erosion. Eur. J.

Soil Sci. 2001, 52, 59–68.

10. Ruiz-Sinoga, J.D.; García-Marín, R.; Martínez-Murillo, J.F.; Gabarrón-Galeote, M.A.

Precipitation dynamics in southern Spain: trends and cycles. Int. J. Climatol. 2010, 31, 2281–2289,

doi:10.1002/joc.2235.

11. Ruiz-Sinoga, J.D.; Martínez-Murillo, J.F.; Gabarrón-Galeote, M.A.; García-Marín, R. Effects of

exposure, scrub position, and soil surface components on the hydrological response in small plots

in southern Spain. Ecohydrology 2010, 3, 402–412, doi:10.1002/eco.159.

12. Gaskin, G.J.; Miller, J.D. Measurement of soil water content using a simplified impedance

measuring technique. J. Agric. Eng. Res. 1996, 63, 153–159.

13. Singh, P.; Kanwar, R.S.; Thompson, M.L. Measurement and characterization of macropores by

using AUTOCAD and automatic image analysis. J. Environ. Qual. 1991, 20, 289–294.

14. Romkens, M.J.M.; Prasad, S.N.; Gerits, J.J.P. Soil erosion modes of sealing soils: a

phenomenological study. Soil Technol. 1997, 11, 31–41.

Page 17: 2015 OPEN ACCESS applied sciences...4 School of Aeronautics & Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China; E-Mail: xianweich@gmail.com 5 School of Science,

Appl. Sci. 2015, 5 173

15. Le Bissonnais, Y.; Bruand, A.; Jamagne, M. Laboratory experimental study of soil crusting:

relation between aggregate breakdown mechanisms and crust structure. Catena 1989, 16, 377–392.

16. Andrea, R.; Helming, K.; Diestel, H. Effect of antecedent soil water content and rainfall regime

on microrelief changes. Soil Technol. 1997, 10, 69–81.

17. Backus, D.; Kehoe, P.J.; Kydland, F.E. Dynamics of the Trade Balance and the Terms of Trade:

The S-curve; National Bureau of Economic Research: Cambridge, MA, USA, 1992.

18. Bejan, A.; Lorente, S. The constructal law origin of the logistics S curve. J. Appl. Phys. 2011, 110,

024901, doi:10.1063/1.3599857.

19. Sun, W.; Won, S.H.; Ju, Y. In situ plasma activated low temperature chemistry and the S-curve

transition in DME/oxygen/helium mixture. Combust. Flame 2014, 161, 2054–2063.

20. De Massis, A.; Chirico, F.; Kotlar, J.; Naldi, L. The temporal evolution of proactiveness in family

firms: The horizontal S-curve hypothesis. Fam. Bus. Rev. 2013, doi:10.1177/0894486513506114.

21. Van Rompaey, A.; Krasa, J.; Dostal, T.; Govers, G. Modelling sediment supply to rivers and

reservoirs in Eastern Europe during and after collectivisation period. Hydrobiologia 2003, 494,

169–176.

22. USDA–Water Environment Prediction Project: Hillslope Profile and Watershed Model

Documentation; Flanagan, D.C.; Nearing, M.A., Eds.; US Department of Agriculture–

Agricultural Research Service, National Soil Erosion Research Laboratory: West Lafayette, IN,

USA 1995.

23. Bathurst, J.C.; Wicks, J.M.; O’Connell, P.E. The SHE/SHESED basin scale water flow and

sediment transport modelling system. In Computer Models of Watershed Hydrology, Singh, V.P.,

Ed.; Water Resource Publication: Highlands Ranch, CO, USA, 1995; pp. 563–594.

24. Hamilton, J.D. Time Series Analysis; Princeton University Press: Princeton, NJ, USA, 1994;

Volume 2.

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article

distributed under the terms and conditions of the Creative Commons Attribution license

(http://creativecommons.org/licenses/by/4.0/).


Recommended